Joint Probability Mass Functions

Hi, I'm revising for an upcoming test and have a past paper with answers but I can't understand where they are coming from. If anyone could try explaining this too me it would be greatly appreciated.

The number of eggs laid by an insect is a random variable which has a poisson distribution with expectation so for Independently for each egg laid there is probability each that it develops into a male insect or a female insect. Let denote the number of male offspring, and the number of female offspring.

i) Find the joint probability mass function of

ii) Show that the marginal distribution of , the number of male offspring, is Poisson with expectation

The answers I've been given are

i) for non negative intergers

ii)For non-negative integers , as the sum is

If anyone can explain how you get to these answers it would be a massive help,

Hi, I'm revising for an upcoming test and have a past paper with answers but I can't understand where they are coming from. If anyone could try explaining this too me it would be greatly appreciated.

The number of eggs laid by an insect is a random variable which has a poisson distribution with expectation so for Independently for each egg laid there is probability each that it develops into a male insect or a female insect. Let denote the number of male offspring, and the number of female offspring.

i) Find the joint probability mass function of

ii) Show that the marginal distribution of , the number of male offspring, is Poisson with expectation

The answers I've been given are

i) for non negative intergers

ii)For non-negative integers , as the sum is

If anyone can explain how you get to these answers it would be a massive help,

Cheers

The number of males has a Poisson distribution with mean , as does the number of females born.